[Environment: Sandbox] Sign-in Message - does anyone know how to stop this popping up?

I've been having this issue since upgrading to iOS 6 on my iPhone 4S back in September or October of 2012. Every so often, usually just after switching networks from Wifi to 3G or launching an app in the morning, I'll get an endless loop of these annoying popups. If I hit cancel it pops up again instantly (or sometimes with maybe a half-second delay). If I hit Continue, I get a username/password dialog. If I enter legit credentials, I get the first dialog box (the "Environment Sandbox" one again, just as though I'd hit cancel). If I enter bad credentials, it complains and then instantly pops up the "Environment Sandbox" dialog again.

So basically, endless dialog box spam. It makes the device completely unusable while it's happening. I find that if I mash the cancel button fast enough, I can get a swipe or a touch in, and I can eventually "crawl" into the Settings app and turn on Airplane Mode, which stops them from popping up.

If I stay in airplane mode for 5-15 minutes, when I leave airplane mode it usually doesn't happen again for a couple hours. The only way I've found to make it go away completely is to stay logged out of my Apple ID.

If I'm logged into my Apple ID, the issue persists through:

1. Rebooting the device. (It typically happens again within 30 seconds).

2. Going into Airplane mode for 15 minutes, and signing out of my Apple ID. Once I go out of airplane mode it doesn't pop up again until I sign back in with my Apple ID. After that, if I log back into my apple ID, (typically through buying something on iTunes), it'll usually happen again within a day.

3. Changing my Apple ID password (I got about 2.5 days of peace and then it happened again)

4. Wiping the device. Once I log in with my Apple ID it starts happening again.

So far Apple's response to my support requests has been ameteurish. The Genius bar flailed helplessly and told me that since it says "Sandbox", that makes it a Developer Issue and they couldn't help, and to go talk to developer support. Apple Developer Support told me that it was a regular device support issue and very rudely told me that they don't cover that kind of stuff, so go to talk to the Genius Bar, and when I told them that I already had, they asked me to file a bug report, which I did but they don't let you link to it.

Eventually I figured out how to avoid the issue by staying logged out of my Apple ID, and decided to talk to Apple ID support, who told me that there was nothing wrong/out of the ordinary going on with my Apple ID, and to try resetting the password. That made the problem go away for 2-3 days and now it's popped up again.